We illustrate our methods for color space embeddings on two examples.
In computer graphics, surfaces are often subdivided into patches that are, e.g., useful in refining a mesh topology or result from a scan of a 3D
object captured from various angles. In the latter case, these parts have to be merged to form a complete 3D model of the object and it is often necessary to manually adjust and correct overlapping parts. Figure6.7shows an example of a 3D model with several patches that are put together. For manual corrections, it is important that adjacent patches can be distinguished well. This problem can be cast into a problem of color space embeddings.
Figure 6.7shows a graphGmodeling the adjacencies in the 3D model, where each vertex represents a patch and there is an edge between two vertices if their corresponding patches are adjacent. Note that G has some edge crossings because of patches on the left hand side of the rabbit. Using an MDS initialization and refinement in Luv, the colors given in Fig. 6.7 are obtained.
Figure 6.5: Laplace matrix initialization in Lab space and Voronoi refinement (lhs) of the original example graph from [38] (rhs)
Figure 6.6: Corresponding positions in Lab space of the lhs of Fig. 6.5
Figure 6.7: Patches colored with refined colors from MDS-initialization in Luv color space and its corresponding patch graph G
A different scenario arises from colored scatterplots. Figure 6.8 shows a scatterplot of data from character recognition, where each vertex represents a hand-written digit (from 1 to 9) and its position is determined from extracted features that shall discriminate the classes, i.e., all vertices corresponding to a certain digit should be close to each other, but well separated from other digits. Since this cannot be achieved in a low-dimensional space (here, it is 3-dimensional space), the clusters necessarily overlap. It is therefore difficult to distinguish between pairs of clusters if their colors are too similar. By creating a weighted graph G, where each vertex represents a cluster and defining weights by a normalized function, monotone in the overlap, the problem is transformed into the kind considered here. We computed the barycenter bj and the standard deviationσj of the distances from bj for each cluster to define weights by
ω({vj, vk}) =
(0 if |bj−bk|> σj +σk , 1 if |bj−bk|<|σj −σk| ,
and linear interpolation otherwise. The result shown in Fig. 6.8 is obtained using the Laplacian matrix for initialization and refinement in Lab space.
Figure 6.8: 3D-scatterplot with Laplace matrix/force-directed Lab-colors
6.6 Conclusion
We presented refined methods for assigning visually distinct colors to vertices of graphs that model the adjacency of geometric objects as, e.g., in the two applications sketched above. These methods naturally apply to other scenarios, such as network analyses in which the output consists of a graph clustered into cohesive or structurally similar groups [12].
Assuming that the visualization is ultimately viewed on a device using RGB or CMYK let us focus on these color spaces. Note that, using the corresponding transformation between the color spaces, Algorithm 9almost immediately works for CMYK. Our methods apply to subsets of color spaces as well, though. For example, when luminosity should be fixed, such that all colors appear roughly of the same intensity, or restrictions to a subset of colors are desired that are well distinguishable for certain forms of color-blindness, this can be simply achieved by, e.g., using only a two-dimensional initialization and an appropriate clipping. Finally, rotating, e.g., the initial Laplacian layout, such that a maximum eigenvector aligns with the black-white axis, and slightly modifying the force-directed refinement, such that forces along the black-white axis are stressed, it is possible to assign colors that are well distinguishable on both color and black-and-white printers.
Summary
This concludes Part II of this work. In Chaps.3to 6we have presented four new applications that use the Laplacian matrixL. Finally, we want to briefly sum up, how concepts of Part I entered PartII.
Chapter 3. Network analysis. The Laplacian matrix is used to com-pute effective resistance Reff(s, t) and throughput τ(v) in an electrical net-work G= (V, E, c) that are needed for the centrality measures current-flow betweenness cCB(v) and current-flow closeness cCC(v).
• Us,t solves the potential equation (on graphs)LUs,t =ρs,t,
• Us,t is a fundamental solution,
• Us,t isharmonic on V \ {s, t},
• Us,t with (LUs,t)s,t =±1 has minimum energy E(Us,t) (Dirichlet’s principle).
Chapter4. Dynamic Graph Drawing. Orthonormal eigenvectorsu2(t), u3(t), . . ., of linearly interpolated Laplacian matrices L(t) are used for dy-namic graph drawing.
• u2(t) is an eigenvector of L(t),
• u2(t) with u2(t)⊥u1(t) and |u2(t)|= 1 has minimum energy E(u2(t)) (Dirichlet’s principle).
Chapter5. Routing. The Laplacian matrix is used to compute the barycen-tric embedding pC = (x, y) of a 3-connected graph.
• x, y solve the two corresponding Dirichlet problems,
• pC with pC|C fixed has minimum energy E(pC) (Dirichlet’s principle),
• positions of inner vertices are within C (maximum principle),
• positions of inner vertices are the barycenter of their neighbors (meanvalue property).
Chapter 6. Coloring. Eigenvector un is used for an embedding χ into a color space C, such that edges are as long as possible, i.e., such that colors are perceived as differently as possible.
• un is an eigenvector of L,
• un with |un|= 1 has maximumenergy E(un).
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A.1 Basic Theorems
This section is a collection of standard theorems that are frequently used.
Theorem A.1.1 (Rayleigh’s inequality) Let M ∈ Rn×n be a positive semi-definite matrix with eigenvalues 0 ≤ λ1 ≤ · · · ≤ λn and correspond-ing eigenvectors u1, . . . , un. For every 06=x∈span{uk, . . . , un},1≤k≤n
λk ≤ xTM x xTx with equality iff x is an eigenvector to λk.
Note that this is also true for the self-adjoint operator ∆ acting on L2(Ω) of a bounded domain Ω with, say, ∂Ω∈ C2,α.
Theorem A.1.2 Let Q ∈ Rn×n be an orthogonal matrix, i.e., Q−1 = QT. Rows (and columns) of Q form an orthonormal basis of Rn, detQ = ±1.
For any symmetric matrix M ∈Rn×n there is an orthogonal matrix Q, such that QTM Q is diagonal. The diagonal contains the eigenvalues of M, and the columns of Q are corresponding unit eigenvectors of M.
Theorem A.1.3 (Gershgorin) Let M ∈ Rn×n. All eigenvalues are con-tained in the union of the disks {z ∈C : |z−Mj,j| ≤P
k6=j|Mj,k|}.
Theorem A.1.4 (Hurwitz) Let M ∈ Rn×n and let Mfk denote the matrix obtained from M by omitting first k rows and columns. The matrix M is positive definite iff detMfk>0 for all 0≤k < n.
Theorem A.1.5 Let F be an R-vector space, and F0 denote all linear func-tions from F to R. If f0f = 0 for all f0 ∈ F0 then f = 0.
A set Ω ∈ Rn is called simply connected (with respect to the canonical topology T induced by the Euclidean norm) if for all x1, x2 ∈ Ω there is a continuous function γ: [0,1] −→ Ω with γ(0) = x1 and γ(1) = x2, and all such functions γ can continuously be transformed into each other.
Theorem A.1.6 Let F: Ω−→ Rn be continuously differentiable. Let Ω be simply connected and ∂jFk(x) = ∂kFj(x) for all x ∈ Ω and 1 ≤ j, k ≤ n.
There is a function U ∈ C1(Ω), such that F =−∇U.
For n = 3 the requirement on the partial derivatives is rot F = 0. If γ: [0,1]−→Ω is a closed continuously differentiable curve
1
Given a set of n vertices v ∈ V and pairwise distances d(v, w), it is easy to define an embedding p: V −→ Rn−1, such that |p(v)−p(w)| = d(v, w).
For embeddings into spaces with lower dimension, this might not be possible exactly. Multidimensional Scaling (MDS) denotes a set of techniques to derive an embedding p into Rk, k < n, such that |p(v)−p(w)| ≈ d(v, w) is fulfilled as well as possible in some sense. For details see, e.g., [7, 29]. First, define the symmetric matrix ˆB by
Bˆv,w :=−1 where µj are eigenvalues of ˆB corresponding to orthonormal eigenvectors yj
and |µn| ≥ |µn−1| ≥ · · · ≥ |µ1|. We just state the following lemma, inspired by [51], that provides a connection between MDS and electrical networks.
Lemma A.2.1 Let G= (V, E, c) be an electrical network and define Bˆ wrt d(v, w)2 := Reff(v, w). The eigenvectors of the (weighted) Laplacian matrix L=L(G) and Bˆ coincide.
Proof: For a given electrical networkG, solutionsU of the potential equation LU =ρ are only unique modulo1. In Sect.3.2, we defined unique solutions